A rigorous path integral for quantum spin using flat-space Wiener regularization

Abstract

Adapting ideas of Daubechies and Klauder [J. Math. Phys. {\bf 26} (1985) 2239] we derive a rigorous continuum path-integral formula for the semigroup generated by a spin Hamiltonian. More precisely, we use spin-coherent vectors parametrized by complex numbers to relate the coherent representation of this semigroup to a suitable Schrödinger semigroup on the Hilbert space $L^2(R^2)$ of Lebesgue square-integrable functions on the Euclidean plane $R^2$. The path-integral formula emerges from the standard Feynman-Kac-Itô formula for the Schrödinger semigroup in the ultra-diffusive limit of the underlying Brownian bridge on $R^2$. In a similar vein, a path-integral formula can be constructed for the coherent representation of the unitary time evolution generated by the spin Hamiltonian.

revised version

References in corpus (1)

• On the Path Integral Representation for Spin Systems

Cited by in corpus (3)

• Semiclassical dynamics of a spin-1/2 in an arbitrary magnetic field
• A lower bound for the Wehrl entropy of quantum spin with sharp high-spin asymptotics
• Construction of Self-Adjoint Berezin-Toeplitz Operators on Kahler Manifolds and a Probabilistic Representation of the Associated Semigroups